CBSE NOTES CLASS 11 PHYSICS CHAPTER 4

MOTION IN A PLANE

VECTOR ALGEBRA

Scalar quantity

A quantity which has only magnitude, but no direction, is called scalar quantity. It is specified completely by a single number, along with the proper unit. Examples - distance, mass, temperature, work, energy, time etc.

Scalars are combined using the rules of ordinary algebra. They can be added, subtracted, multiplied and divided just as the ordinary numbers.

Vector quantity

A quantity, that has both a magnitude and a direction and obeys the triangle law of addition or equivalently the parallelogram law of addition, is called vector quantity. A vector is specified by giving its magnitude by a number and its direction together with appropriate unit. Examples - displacement, velocity, acceleration and force, torque, momentum, angular velocity etc.

Representation of a vector

When printed, a vector can be represented by bold face, like v, F etc. or it can be represented by an arrow placed over a letter, say $\stackrel{\to }{\mathrm{v}}$. Graphically the vector is represented by a ray originating at the initial position and terminating at the final position and an arrow marked at the final position.

The magnitude of a vector is called its absolute value indicated by |$\stackrel{\to }{\mathrm{v}}$| or v.

Position vector

If O is the origin of the Cartesian coordinate system and P is the position of a point at time t, then the vector representing the straight line from O to P, $\stackrel{\to }{\mathrm{O}\mathrm{P}}$, is called the position vector of the point at time t and is represented by $\stackrel{\to }{\mathrm{r}}$. Similarly point P' can be represented by position vector $\stackrel{\to }{\mathrm{r}\mathrm{\text{'}}}$.

Displacement Vector

Displacement vector is the straight line joining the initial and final positions, eg. , . It depends only on the initial and final positions of the object and does not depend on the actual path undertaken by the object between the two positions.

Equality of Vectors

Two vectors $\stackrel{\to }{\mathrm{A}}$ and $\stackrel{\to }{\mathrm{B}}$ are said to be equal if and only if they have the same magnitude and the same direction.

Multiplication of Vectors by Real Numbers

Multiplying a vector $\stackrel{\to }{\mathrm{A}}$ with a positive number λ gives a vector whose magnitude is changed by the factor λ but the direction is the same as that of $\stackrel{\to }{\mathrm{A}}$

= if λ > 0.

Multiplying a vector $\stackrel{\to }{\mathrm{A}}$ by a negative number λ gives a vector whose direction is opposite to the direction of $\stackrel{\to }{\mathrm{A}}$ and whose magnitude is .

Triangle law of vector addition states that if two vectors are represented by two sides of a triangle (in direction and in magnitude), then the resultant vector is represented by the third side of the triangle (in direction and in magnitude).

The difference of two vectors $\stackrel{\to }{\mathrm{A}}$ and $\stackrel{\to }{\mathrm{B}}$ can be represented as the sum of two vectors $\stackrel{\to }{\mathrm{A}}$ and $\stackrel{\to }{-\mathrm{B}}$

• Only vectors of same nature can be added.

• The addition of two vector $\stackrel{\to }{\mathrm{A}}$ and $\stackrel{\to }{\mathrm{B}}$ gives resultant $\stackrel{\to }{\mathrm{R}}$

Parallelogram Law of Vector Addition states that when two vectors are represented by two adjacent sides of a parallelogram in direction and magnitude then the resultant of these vectors is represented in magnitude and direction by the diagonal of the parallelogram starting from the same point.

And tan β =

• Vector addition is commutative $\stackrel{\to }{\mathrm{A}}$+$\stackrel{\to }{\mathrm{B}}$ = $\stackrel{\to }{\mathrm{B}}$+$\stackrel{\to }{\mathrm{A}}$

• Vector addition is associative, A + ($\stackrel{\to }{\mathrm{B}}$+C) = ($\stackrel{\to }{\mathrm{A}}$+$\stackrel{\to }{\mathrm{B}}$) + C

• R is maximum if θ = 0 and minimum if θ = 180o

Where, θ is the angle between vector $\stackrel{\to }{\mathrm{A}}$ and vector $\stackrel{\to }{\mathrm{B}}$, and β is the angle which R makes with the direction of $\stackrel{\to }{\mathrm{A}}$.

Proof of Parallelogram Law

Let $\stackrel{\to }{\mathrm{O}\mathrm{P}}$ and $\stackrel{\to }{\mathrm{O}\mathrm{Q}}$ represent the two vectors $\stackrel{\to }{\mathrm{A}}$ and $\stackrel{\to }{\mathrm{B}}$ making an angle θ with each other. $\stackrel{\to }{\mathrm{O}\mathrm{S}}$ represents the resultant vector $\stackrel{\to }{\mathrm{R}}$.

SN is normal to OP and PM is normal to OS.

From the geometry of the figure,

OS2 = ON2 + SN2

but ON = OP + PN = A + B cos θ

SN = B sin θ

OS2 = (A + B cos θ)2 + (B sin θ)2

or, R2 = A2 + B2 + 2AB cos θ

(Law of cosines)

Now,

In Δ OSN, SN = OS sin β = R sin β, and

in Δ PSN, SN = PS sin θ = B sin θ

Therefore, R sin β = B sin θ

Similarly, PM = A sin β = B sin α

This is called law of sines.

Null Vector

A vector having zero magnitude is called null vector or a zero vector and is represented by 0.

Unit Vector

A unit vector is a vector of unit magnitude and points in a particular direction. It has no dimension and unit. It is used to specify a direction only. Unit vectors along the x-, y- and z-axes of a rectangular coordinate system are denoted by $\stackrel{̂}{\mathrm{i}}$, $\stackrel{̂}{\mathrm{j}}$ and $\stackrel{̂}{\mathrm{k}}$, respectively.

Since these are unit vectors, we have

|$\stackrel{̂}{\mathrm{i}}$| = |$\stackrel{̂}{\mathrm{j}}$| = |$\stackrel{̂}{\mathrm{k}}$| = 1

Rectangular components of a vector in a plane

A general vector can be resolved into components along the axes of a rectangular coordinate system. The origin and coordinate system can be arbitrarily chosen as per convenience.

If vector A makes an angle θ with x-axis and Ax and Ay are the rectangular components of $\stackrel{\to }{\mathrm{A}}$ along X- and Y-axis respectively. Then

= Ax $\stackrel{̂}{\mathrm{i}}$+ Ay $\stackrel{̂}{\mathrm{j}}$

A2 = Ax2 + Ay2

tan θ = $\frac{{\mathrm{B}}_{\mathrm{y}}}{{\mathrm{A}}_{\mathrm{x}}}$

Where, Ax =A cos θ and By = A sin θ

If α, β, and γ are the angles which A makes with the x-, y-, and z-axes, respectively then,

= Ax $\stackrel{̂}{\mathrm{i}}$+ Ay $\stackrel{̂}{\mathrm{j}}$ + Az $\stackrel{̂}{\mathrm{k}}$

A2 = Ax2 + Ay2 + Az2

If = Ax $\stackrel{̂}{\mathrm{i}}$+ Ay $\stackrel{̂}{\mathrm{j}}$ + Az $\stackrel{̂}{\mathrm{k}}$

And = Bx $\stackrel{̂}{\mathrm{i}}$+ By $\stackrel{̂}{\mathrm{j}}$ + Bz $\stackrel{̂}{\mathrm{k}}$

Then $\stackrel{\to }{\mathrm{R}}$ = $\stackrel{\to }{\mathrm{A}}$ + $\stackrel{\to }{\mathrm{B}}$

= Rx $\stackrel{̂}{\mathrm{i}}$ + Ry $\stackrel{̂}{\mathrm{j}}$ + Rz $\stackrel{̂}{\mathrm{k}}$

Where Rx = Ax + Bx,

Ry = Ay + By

and Rz = Az + Bz

Dot product or scalar product

The dot product of two vectors $\stackrel{\to }{\mathrm{A}}$ and $\stackrel{\to }{\mathrm{B}}$, represented by ‘.’ is a scalar, which is equal to the product of the magnitudes of $\stackrel{\to }{\mathrm{A}}$ and $\stackrel{\to }{\mathrm{B}}\mathbf{}$and the cosine of the smaller angle between them. If θ is the smaller angle between $\stackrel{\to }{\mathrm{A}}$ and $\stackrel{\to }{\mathrm{B}}$, then $\stackrel{\to }{\mathrm{A}}$.$\stackrel{\to }{\mathrm{B}}$= AB cos θ

$\stackrel{\to }{\mathrm{A}}$.$\stackrel{\to }{\mathrm{B}}$ = $\stackrel{\to }{\mathrm{B}}$.$\stackrel{\to }{\mathrm{A}}$

If = Ax $\stackrel{̂}{\mathrm{i}}$+ Ay $\stackrel{̂}{\mathrm{j}}$ + Az $\stackrel{̂}{\mathrm{k}}$

And = Bx $\stackrel{̂}{\mathrm{i}}$+ By $\stackrel{̂}{\mathrm{j}}$ + Bz $\stackrel{̂}{\mathrm{k}}$

Then, $\stackrel{\to }{\mathrm{A}}$.$\stackrel{\to }{\mathrm{B}}$ = Ax Bx + Ay By + Az Bz

Projection of $\stackrel{\to }{\mathrm{A}}$ on $\stackrel{\to }{\mathrm{B}}$ = $\frac{\stackrel{\to }{\mathrm{A}}.\stackrel{\to }{\mathrm{B}}}{\mathrm{A}}$ = A cos θ

cos θ = $\frac{\stackrel{\to }{\mathrm{A}}.\mathbit{}\stackrel{\to }{\mathrm{B}}}{\mathrm{A}\mathrm{B}}$

• Scalar product is zero if $\stackrel{\to }{\mathrm{A}}$ and $\stackrel{\to }{\mathrm{B}}$ are perpendicular to each other.

$\stackrel{̂}{\mathrm{i}}$.$\stackrel{̂}{\mathrm{i}}$ = $\stackrel{̂}{\mathrm{j}}$.$\stackrel{̂}{\mathrm{j}}$ = $\stackrel{̂}{\mathrm{k}}$.$\stackrel{̂}{\mathrm{k}}$ = 1

$\stackrel{̂}{\mathrm{i}}$.$\stackrel{̂}{\mathrm{j}}$ = $\stackrel{̂}{\mathrm{j}}$.$\stackrel{̂}{\mathrm{k}}$ = $\stackrel{̂}{\mathrm{k}}$.$\stackrel{̂}{\mathrm{i}}$ = 0

If $\stackrel{\to }{\mathrm{A}}$.$\stackrel{\to }{\mathrm{B}}$ = 0, either $\stackrel{\to }{\mathrm{A}}$ = 0 or $\stackrel{\to }{\mathrm{B}}$ = 0 or θ = 90o

Cross or Vector product

The cross product of two vectors and, represented by × is a vector, whose magnitude is equal to the product of the magnitudes of $\stackrel{\to }{\mathrm{A}}$ and $\stackrel{\to }{\mathrm{B}}$ and the sine of the smaller angle between them and is perpendicular to both $\stackrel{\to }{\mathrm{A}}$ and $\stackrel{\to }{\mathrm{B}}$. The direction can be found by right hand rule. If θ is the smaller angle between A and B, then

× $\stackrel{\to }{\mathrm{B}}$ = AB Sin θ $\stackrel{̂}{\mathrm{n}}$

Here $\stackrel{̂}{\mathrm{n}}$ is the unit vector perpendicular to the plane of both $\stackrel{\to }{\mathrm{A}}$ and $\stackrel{\to }{\mathrm{B}}$

× $\stackrel{\to }{\mathrm{B}}$ = - × $\stackrel{\to }{\mathrm{A}}$

$\stackrel{̂}{\mathrm{i}}$ × $\stackrel{̂}{\mathrm{i}}$ = $\stackrel{̂}{\mathrm{j}}$ × $\stackrel{̂}{\mathrm{j}}$ = $\stackrel{̂}{\mathrm{k}}$ × $\stackrel{̂}{\mathrm{k}}$ = 0,

$\stackrel{̂}{\mathrm{i}}$ × $\stackrel{̂}{\mathrm{j}}$ = $\stackrel{̂}{\mathrm{k}}$, $\stackrel{̂}{\mathrm{j}}$ × $\stackrel{̂}{\mathrm{k}}$ = $\stackrel{̂}{\mathrm{i}}$, $\stackrel{̂}{\mathrm{k}}$ × $\stackrel{̂}{\mathrm{i}}$ = $\stackrel{̂}{\mathrm{j}}$

× $\stackrel{\to }{\mathrm{B}}$ = $\left|\begin{array}{ccc}\stackrel{̂}{\mathrm{i}}& \stackrel{̂}{\mathrm{j}}& \stackrel{̂}{\mathrm{k}}\\ {\mathrm{A}}_{\mathrm{x}}& {\mathrm{A}}_{\mathrm{y}}& {\mathrm{A}}_{\mathrm{z}}\\ {\mathrm{B}}_{\mathrm{x}}& {\mathrm{B}}_{\mathrm{y}}& {\mathrm{B}}_{\mathrm{z}}\end{array}\right|$

• Vector product is zero if $\stackrel{\to }{\mathrm{A}}$ and $\stackrel{\to }{\mathrm{B}}$ are parallel to each other.

• It is maximum when θ = 90o

• If $\stackrel{\to }{\mathrm{A}}\mathbf{}$× $\stackrel{\to }{\mathrm{B}}$ = 0, either $\stackrel{\to }{\mathrm{A}}$ = 0 or $\stackrel{\to }{\mathrm{B}}$ = 0 or θ = 0

MOTION IN A PLANE

Position Vector and Displacement

The position vector $\stackrel{\to }{\mathrm{r}}$ of a particle P located in a plane with reference to the origin of an x-y reference frame is given by

$\stackrel{\to }{\mathrm{r}}$ = x $\stackrel{̂}{\mathrm{i}}$+ y $\stackrel{̂}{\mathrm{j}}$

Where x and y are components of $\stackrel{\to }{\mathrm{r}}$ along x-, and y- axes, respectively. They are simply the coordinates of the object.

Let us consider a particle moving along the curve and is at P1 at time t1 and P2 at time t2. Then, the displacement is given by

and is directed from P1 to P2.

$\mathrm{\Delta }\stackrel{\to }{\mathrm{r}}$ = (x2 $\stackrel{̂}{\mathrm{i}}$+ y2 $\stackrel{̂}{\mathrm{j}}$) − (x1 $\stackrel{̂}{\mathrm{i}}$ + y1 $\stackrel{̂}{\mathrm{j}}$)

= $\stackrel{̂}{\mathrm{i}}$ Δx + $\stackrel{̂}{\mathrm{j}}$ Δy

where Δx = x2x1, Δy = y2y1

Velocity Vector

The average velocity $\left(\stackrel{\to }{\mathrm{v}}\right)$ of an object is the ratio of the displacement and the corresponding time interval

= vx $\stackrel{̂}{\mathrm{i}}$ + vy $\stackrel{̂}{\mathrm{j}}$

Direction of $\stackrel{\to }{\mathrm{v}}$ is the same as $\mathrm{\Delta }\stackrel{\to }{\mathrm{r}}$ (displacement).

Instantaneous velocity $\stackrel{\to }{\mathrm{v}}$ is given by the limiting value of the average velocity as the time interval approaches zero

The direction of velocity at any point on the path of an object is tangential to the path at that point and is in the direction of motion.

where θ is the angle made by the velocity vector with x-axis.

Acceleration Vector

The average acceleration a of an object moving in x-y plane for a time interval Δt is the change in velocity divided by the time interval

$\stackrel{‾}{\mathrm{a}}$ = ax $\stackrel{̂}{\mathrm{i}}$ + ay $\stackrel{̂}{\mathrm{j}}$

Instantaneous acceleration $\stackrel{\to }{\mathbf{a}}$ is given by the limiting value of the average acceleration as the time interval approaches zero

Where,

and

• In one dimension, the velocity and the acceleration of an object are always along the same straight line (either in the same direction or in the opposite direction). However, for motion in two or three dimensions, velocity and acceleration vectors may have any angle between 0° and 180° between them.

MOTION IN A PLANE WITH CONSTANT ACCELERATION

Motion in a plane (two-dimensions) can be treated as two separate simultaneous one-dimensional motions with constant acceleration along two perpendicular directions.

Equations of motion can be written as

• vx = ux + axt

and vy = uy + ayt

• x = xo + uxt + ½ axt2

and y = yo + uyt + ½ ayt2

• vx2 - ux2 = 2ax (x - xo)

and vy2 – uy2 = 2ay (y - yo)

• Relative velocity of object A relative to that of B is $\stackrel{\to }{\mathrm{v}}$AB = $\stackrel{\to }{\mathrm{v}}$A$\stackrel{\to }{\mathrm{v}}$B (vectors)

Relative velocity of object B relative to that of A is $\stackrel{\to }{\mathrm{v}}$BA = $\stackrel{\to }{\mathrm{v}}$B$\stackrel{\to }{\mathrm{v}}$A

Therefore, $\stackrel{\to }{\mathrm{v}}$AB = – $\stackrel{\to }{\mathrm{v}}$BA and |$\stackrel{\to }{\mathrm{v}}$AB| = |$\stackrel{\to }{\mathrm{v}}$BA|

Projectile Motion

An object that is in flight after being thrown or projected is called a projectile. For example football, a cricket ball, a baseball etc.

The motion of a projectile may be considered as the result of two separate, simultaneously occurring components of motions, in horizontal and vertical direction respectively.

Assume that the air resistance has negligible effect on the motion of the projectile.

Let us consider a projectile launched with initial velocity u that makes an angle θ with the x-axis.

ax = 0, ay = –g

ux = u cos θ, uy = u sin θ

If we take the initial position to be the origin of the reference frame, we have xo = 0, yo = 0

Since there is no acceleration in the horizontal direction,

vx = ux = u cos θ = constant,

vy = uy - gt = u sin θ - gt

Also x = ux t = (u cos θ) t --- (0)

y = (u sin θ) t – $\frac{1}{2}$ gt2 --- (1)

Time at the maximum height

At the point of maximum height, vy = 0 and therefore,

⇒ vy = 0

u sin θ - g tm = 0

tm = --- (2)

Maximum height

Now putting the value of t in equation (1), we have,

Relation between x and y (Equation of projectile)

Eliminating t from equation (0) and (1),

This equation is of the form y = ax + bx2, in which a and b are constants. This is the equation of a parabola, i.e. the path of the projectile is a parabola.

Time of flight

Putting y = 0 in equation (1),

Horizontal range

Maximum distance travelled by the projectile is called the horizontal range R. Since ux is constant throughout the flight,

R is maximum, when θ = $\frac{\mathrm{\pi }}{4}$

${⇒\mathrm{R}}_{\mathrm{m}}={\mathrm{u}}^{2}/\mathrm{g}$

Uniform Circular Motion

When an object follows a circular path at constant speed, the motion of the object is called uniform circular motion.

The magnitude of velocity is constant, but the direction is constantly changing such that the velocity is always tangential to the circular path or perpendicular to the radius vector.

Angular Displacement

Angle swept by the radius vector of a particle moving on a circular path is known as angular displacement of the particle.

Centripetal Acceleration

Let $\stackrel{\to }{{\mathrm{r}}_{1}}$ and $\stackrel{\to }{{\mathrm{r}}_{2}}$ be the position vectors and $\stackrel{\to }{{\mathrm{v}}_{1}}$ and $\stackrel{\to }{{\mathrm{v}}_{2}}$ be the velocity vectors at points points P1 and P2 respectively

Δr and Δv are perpendicular to each other or Δv is directed towards the centre, hence

will also be directed towards the centre of the circle.

Now Δ ABC and Δ OP1P2 are similar. Hence

ac is called centripetal acceleration as it is directed towards the centre.

Angular Acceleration (α)

Now,

And,

Therefore,

Hence ac = ω2r

Also v =

And f =

$⇒$ ac = 4

Now,

Tangential Acceleration

If the speed is not constant, then there is also a tangential acceleration at. The tangential acceleration is tangent to the path of the particle's motion or in the direction of velocity.

Here, at = $\frac{\mathrm{d}\mathrm{v}}{\mathrm{d}\mathrm{t}}$ and ac = ar= $\frac{{\mathrm{v}}^{2}}{\mathrm{r}}$

And a = $\sqrt{{{\mathrm{a}}_{\mathrm{t}}}^{2}+{{\mathrm{a}}_{\mathrm{C}}}^{2}}$